The key difference between QA technology and document search is that document search takes a keyword query and returns a list of documents, ranked in order of relevance to the query (often based on popularity and page ranking), while QA technology takes a question expressed in natural language, seeks to understand it in much greater detail, and returns a precise answer to the question

Elasticsearch is a search and analytics engine. You can use it to search whatever you want: natural text (in any language), images, DNA sequences, etc. Elasticsearch gives you documents that match your query. "Find all documents with text similar to 'quick fox' AND price > 5"

Watson is a "question answering engine" driven by machine learning and natural language processing. Watson answers questions, as opposed to finding documents that are relevant to your query. "What is 5 + 5?"

Parts of Watson internally use search technology. Or put another way, you could build a Watson yourself which utilizes Elasticsearch for some of the processing (searching the expert knowledge, retrieving relevant docs, calculating feature/language vectors, etc), but you would need a lot of other components too (language decomposition, PoS tagging, hypothesis generation/testing, etc)